What is the volume of the square pyramid shown below?
2 answers:
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Answer:
a. Line
b. Plane
c. All of R^3
Step-by-step explanation:
In order to answer this question, we need to study the linear independence between the vectors :
1 - A set of three linearly independent vectors in R^3 generates R^3.
2 - A set of two linearly independent vectors in R^3 generates a plane.
3 - A set of one vector in R^3 generates a line.
The next step to answer this question is to analyze the independence between the vectors of each set. We can do this by putting the vectors into the row of a R^(3x3) matrix. Then, by working out with the matrix we will find how many linearly independent vectors the set has :
a. Let's put the vectors into the rows of a matrix :
⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒
We find that the second vector is a linear combination from the first and the third one (in fact, the second vector is the first vector multiply by -3).
We also find that the third vector is a linear combination from the first and the second one (in fact, the third vector is the first vector multiply by 5).
At the end, we only have one vector in R^3 ⇒ The set of all linear combinations of the set a. is a line in R^3.
b. Again, let's put the vectors into the rows of a matrix :
⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒
We find that there are only two linearly independent vectors in the set so the set of all linear combinations of the set b. is a plane (in fact, the third vector is equivalent to the first vector plus three times the second vector).
c. Finally :
⇒ Applying matrix operations we find that the matrix is equivalent to this another matrix ⇒
The set is linearly independent so the set of all linear combination of the set c. is all of R^3.